Spectrally-efficient spiral-based waveforms for communication

ABSTRACT

Methods for communicating are disclosed. A method includes obtaining at least one input communication symbol selected from a set of communication symbols, converting the at least one input communication symbol into at least one transmittable waveform using at least one defined spiral waveform function, and transmitting the at least one transmittable waveform over a communication channel. Example spiral waveform functions include spline-based piecewise functions and Archimedes spiral functions.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority from U.S. Provisional PatentApplication No. 62/155,856, filed on May 1, 2015, entitled“Spectrally-efficient spiral-based waveforms for communication,” theentire contents of which are hereby incorporated by reference.

BACKGROUND

Applicant's prior patents U.S. Pat. No. 8,472,534 entitled“Telecommunication Signaling Using Non-Linear Functions” and U.S. Pat.No. 8,861,327 entitled “Methods and Systems for Communicating”, thecontents of which are herein incorporated by reference in theirentirety, introduced spiral-based signal modulation. Spiral-based signalmodulation may base signal modulation on complex spirals, rather thanthe traditional complex circles used by standard signal modulationtechniques such as Quadrature Amplitude Modulation (QAM) and Phase-ShiftKeying (PSK).

In spiral modulation, symbol waveforms may be constructed by multiplyinga complex circle, whose period may correspond to the symbol time, by asequence of real amplitude values that may vary over the symbol time. Toavoid boundary amplitude discontinuities, these amplitude values mayfirst increase then decrease over the symbol time, returning to theinitial amplitude. Symbol waveforms may be distinguished from each otherby this amplitude variation, a technique that may be called“intra-symbol amplitude modulation”. Standard modulation techniques,such as phase modulation, may be additionally applied.

Intra-symbol amplitude modulation provides new ways to distinguishbetween symbol waveforms, and therefore to potentially improvecommunication performance. However, certain inefficiencies may still bepresent in existing intra-symbol amplitude modulation-basedcommunication methods, and further refinements to intra-symbol amplitudemodulation-based communication technology may still be made.

SUMMARY

Applicant's prior patents noted above disclosed an exemplaryimplementation of spiral modulation in which spiral-based symbolwaveforms were formed through multiplying a complex circle by a risingexponential (the “head function”) for part of the symbol time,optionally connected to a “tail function” that returned the amplitude toits original value. The intra-symbol amplitude variation was thereforedefined, at least for the “head function” portion of the symbol time, bythe properties of an exponential.

While exponentials have useful mathematical properties, they may not beoptimal for minimizing spectrum usage. According to some exemplaryembodiments, other exemplary techniques for forming spirals with morespectrally-efficient characteristics may be available, and acommunication method making use of intra-symbol amplitude modulation mayincorporate such techniques in order to minimize derivativediscontinuities. Three such exemplary techniques may include twovariations of splines, and the use of linear amplitude variation(generating an “Archimedes spiral”). However, other techniques forgenerating smooth amplitude variations that reduce spectrum usagecompared to exponentials may also be used, as desired.

According to an exemplary embodiment, a method for communicating mayinclude obtaining at least one input communication symbol selected froma set of communication symbols; converting the at least one inputcommunication symbol into at least one transmittable waveform using atleast one non-periodic function selected from a set of non-periodicfunctions; and transmitting the at least one transmittable waveform overa communication channel, wherein at least one non-periodic functionselected from the set of non-periodic functions is a complex circlefunction multiplied by a piecewise function, the piecewise functioncomprising a plurality of cubic piecewise polynomials and having theform

${S(x)} = \left\{ \begin{matrix}{{{p_{1}(x)} = {a_{1} + \;{b_{1}x} + {c_{1}x^{2}} + {d_{1}x^{3}}}},{x \in \left\lbrack {x_{0},x_{1}} \right\rbrack}} \\\vdots \\{{{p_{n}(x)} = {a_{n} + {b_{n}x} + {c_{n}x^{2}} + {d_{n}x^{3}}}},{x \in \left\lbrack {x_{n - 1},x_{n}} \right\rbrack}}\end{matrix} \right.$wherein, for a cubic polynomial p defined over a particular interval, a,b, c, and d are constants defined for that cubic polynomial over theparticular interval.

Another exemplary method for communicating may include obtaining atleast one input communication symbol selected from a set ofcommunication symbols; converting the at least one input communicationsymbol into at least one transmittable waveform using at least onenon-periodic function selected from a set of non-periodic functions; andtransmitting the at least one transmittable waveform over acommunication channel, wherein at least one non-periodic functionselected from the set of non-periodic functions comprises an ArchimedesSpiral function having the form

${f(t)} = {\left( {a + {bt}} \right)e^{\frac{2\;\pi\; j}{C}t}}$wherein a is a constant defining a starting amplitude, b is a slope, Cis a constant defining the rate of angular progression of the spiral incomplex space, and j is an imaginary square root of minus one.

BRIEF DESCRIPTION OF THE FIGURES

Advantages of embodiments of the present invention will be apparent fromthe following detailed description of the exemplary embodiments thereof,which description should be considered in conjunction with theaccompanying drawings in which like numerals indicate like elements, inwhich:

FIG. 1 may show an exemplary plot of a spiral-based waveform and itsspectrum usage.

FIG. 2 may show another exemplary plot of a spiral-based waveform andits spectrum usage.

FIG. 3 may show a further exemplary embodiment of a spiral-basedwaveform and its spectrum usage.

DETAILED DESCRIPTION OF THE EMBODIMENTS

Aspects of the invention are disclosed in the following description andrelated drawings directed to specific embodiments of the invention.Alternate embodiments may be devised without departing from the spiritor the scope of the invention. Additionally, well-known elements ofexemplary embodiments of the invention will not be described in detailor will be omitted so as not to obscure the relevant details of theinvention. Further, to facilitate an understanding of the descriptiondiscussion of several terms used herein follows.

The word “exemplary” is used herein to mean “serving as an example,instance, or illustration.” Any embodiment described herein as“exemplary” is not necessarily to be construed as preferred oradvantageous over other embodiments. Likewise, the term “embodiments ofthe invention” does not require that all embodiments of the inventioninclude the discussed feature, advantage or mode of operation.

Further, many embodiments are described in terms of sequences of actionsto be performed by, for example, elements of a computing device. It willbe recognized that various actions described herein can be performed byspecific circuits (e.g., application specific integrated circuits(ASICs)), by program instructions being executed by one or moreprocessors, or by a combination of both. Additionally, these sequence ofactions described herein can be considered to be embodied entirelywithin any form of computer readable storage medium having storedtherein a corresponding set of computer instructions that upon executionwould cause an associated processor to perform the functionalitydescribed herein. Thus, the various aspects of the invention may beembodied in a number of different forms, all of which have beencontemplated to be within the scope of the claimed subject matter. Inaddition, for each of the embodiments described herein, thecorresponding form of any such embodiments may be described herein as,for example, “logic configured to” perform the described action.

According to an exemplary embodiment, and referring generally to theFigures, various exemplary implementations of spiral modulation in whichspiral-based symbol waveforms may be formed may be disclosed. Suchwaveforms may have utility in, for example, communication using theelectromagnetic spectrum.

According to some exemplary embodiments, multiple potentialimplementations of spiral-based symbol waveforms, and multipletechniques for generating those implementations, may be disclosed.Certain of the implementations or techniques may have particularbenefits when used for particular applications or when used inparticular roles. For example, an exemplary embodiment of a method forcommunicating may use spiral-based symbol waveforms in order to optimizebandwidth usage; according to such an exemplary embodiment, specificspiral-based symbol waveforms may be chosen that optimize bandwidthusage and allow for better bandwidth control. According to anotherexemplary embodiment, an exemplary embodiment of a method forcommunicating may use spiral-based symbol waveforms in order to minimizecoherent interference rejection or power usage, and specificspiral-based symbol waveforms may be chosen to optimize these qualities.Optimizations to other qualities may be made, as desired.

According to an exemplary embodiment disclosed in Applicant's abovepatents, spiral-based symbol waveforms may be implemented by multiplyinga complex circle by a rising exponential (the “head function”) for partof the symbol time; this may optionally be connected to a “tailfunction” that returns the amplitude to its original value. Theintra-symbol amplitude variation of the spiral-based symbol waveform maytherefore be defined, at least for the “head function” portion of thesymbol time, by the properties of an exponential.

While exponentials have useful mathematical properties, they may not beoptimal for minimizing spectrum usage. According to other exemplaryembodiments, other exemplary techniques for forming spirals with morespectrally-efficient characteristics may also be available; for example,this may include techniques for generating smooth amplitude variationsthat reduce spectrum usage compared to exponentials, or othertechniques, as desired.

According to an exemplary embodiment, techniques for generating smoothamplitude variations that reduce spectrum usage compared to exponentialsmay make use of spline interpolation. Though spline interpolation may beunderstood in the art, a brief overview of spline interpolation may beprovided for purposes of clarity.

As understood in the art, spline interpolation may be a form ofinterpolation that makes use of an interpolant called a spline. The“spline” terminology may be derived from a type of elastic ruler thatwas used in shipbuilding; a number of predefined points (“knots”) wouldbe drawn on a grid, and the ruler (the spline) would be bent to passthrough all of the points, thereby defining a curved line. Splineinterpolation may make use of piecewise polynomials, defined by a numberof sub-functions, and which pass through a number of predefined points;the resulting spline curves generally have a curved shape.

Spline interpolation may be understood as follows. We may start with afunction that is tabulated over n+1 points, which we may define asƒ_(k)=ƒ(x_(k)), k=0 . . . n. A spline may be a polynomial constructedbetween each pair of tabulated points, with some degree of non-locality(i.e. the polynomial also depends on other points other than the pair oftabulated points) in order to ensure smoothness along the resultingcurve. Over any particular interval (x_(k), x_(k+1)), we may be able toconstruct a linear function using the following formula:ƒ=Aƒ _(k) +Bƒ _(k+1)  (1)where A is defined as

$\frac{x_{k + 1} - x}{x_{k + 1} - x_{k}}$and B is defined as

$\frac{x - x_{k}}{x_{k + 1} - x_{k}}.$

However, such a function will not be as smooth an approximation as maybe desirable. In order to generate a smooth approximation that iscontinuous in both its first and second derivatives, an alternativenonlinear function may be desirable. For example, it may be possible togenerate a quadratic spline, or a cubic spline.

A cubic spline can be generated for a series of data pointsƒ_(k)=ƒ(x_(k)), k=0 . . . n by adding the tabulated second derivativesof ƒ_(k), as in the following equation:ƒ=Aƒ _(k) +Bƒ _(k+1) +Cƒ _(k) ″+Dƒ _(k+1)″  (2)wherein A and B are defined as above, C is defined as⅙(A³−A)(x_(k+1)−x_(k))², and D is defined as ⅙(B³−B)(x_(k+1)−x_(k))².

By adding in the continuity requirements, and making certainsubstitutions, the following n−1 linear equations can be generated forthe set of n+1 unknowns specified above:

$\begin{matrix}{{{\frac{x_{k} - x_{k + 1}}{6}f_{k - 1}^{''}} + {\frac{x_{k + 1} - x_{k + 1}}{3}f_{k}^{''}} + {\frac{x_{k + 1} - x_{k}}{6}f_{k + 1}^{''}}} = {\frac{f_{k + 1} - f_{k}}{x_{k + 1} - x_{k}} - \frac{f_{k} - f_{k - 1}}{x_{k} - x_{k - 1}}}} & (3)\end{matrix}$

This yields a two-parameter family of possible solutions. For a uniquesolution, such as may be used to generate a spiral waveform, furtherconditions (generally boundary conditions) may be specified.

Once boundary conditions or other conditions are specified, a solutionto this system of equations may be calculated, yielding a piecewisepolynomial S(x) governing the behavior of a spline curve. Thispolynomial may take the form

$\begin{matrix}{{S(x)} = \left\{ \begin{matrix}{{{p_{1}(x)} = {a_{1} + \;{b_{1}x} + {c_{1}x^{2}} + {d_{1}x^{3}}}},{x \in \left\lbrack {x_{0},x_{1}} \right\rbrack}} \\\vdots \\{{{p_{n}(x)} = {a_{n} + {b_{n}x} + {c_{n}x^{2}} + {d_{n}x^{3}}}},{x \in \left\lbrack {x_{n - 1},x_{n}} \right\rbrack}}\end{matrix} \right.} & (4)\end{matrix}$wherein, for a cubic polynomial p_(k) defined over a particular interval(x_(k−1), x_(k)), a_(k), b_(k), c_(k), and d_(k) are constants definedfor that cubic polynomial over that particular interval. The values ofeach of these constants may vary depending on factors like theconditions that are imposed to construct the spline polynomials;different conditions may yield different values for a_(k), b_(k), c_(k),and d_(k).

For example, multiple different types of cubic splines, such as “naturalsplines,” “clamped splines,” or “not-a-knot splines,” may each requirethat different end conditions be specified for a given spline data set.For example, in a “natural spline,” the second derivative of the splinecurve may be defined as being zero at each end. In a “clamped spline,”the first derivative of the spline curve may be specified at each of theends (and may also be zero). A “clamped spline” wherein the firstderivatives at either end are defined to be zero may be referred to as a“zero derivative” spline. Last, in a “not-a-knot” spline, the first andlast end points may have a third derivative that is equal to the thirdderivative of the next outermost point (that is, the second andsecond-to-last points). This will effectively mean that the second andsecond-to-last points are not knots.

For example, according to an exemplary embodiment, a first exemplarytechnique may involve a first variation of splines, and may include a“Not-a-Knot” end condition, wherein the third derivative of the firstpoint in a series of points may be the same as the second point in theseries of points, and wherein the third derivative of the last point inthe series of points may be the same as the second-to-last point in theseries of points. A spiral-based symbol waveform whose intra-symbolamplitude variation is determined by the spline “Not-a-Knot” endcondition may be constructed, for example, from the following exemplaryMATLAB code as would be understood by a person of ordinary skill in theart.

Exemplary MATLAB code for Spline, “Not-a-Knot” end condition:

-   -   min_amp=1; % Minimum complex amplitude    -   max_amp=2; % Maximum complex amplitude    -   numpts=32; % Number of points in symbol waveform    -   x=[1 ceil(numpts/2) numpts]; % Indices for start, mid, end        amplitudes    -   y=[min_amp max_amp min_amp]; % Start, middle, end complex        amplitudes    -   idx=1:numpts; % Indices to interpolate into    -   amps=spline(x, y, idx); % Interpolate amplitudes using spline    -   angles=(idx−1)*2*pi/(numpts−1); % Angles for spiral    -   spiral=amps.*exp(1j*angles); % Spiral=amplitudes.*complex circle

Exemplary FIG. 1 depicts a plot of the resulting spiral-based symbolwaveform and its spectrum usage. A real 102 and an imaginary 104component of the spiral may be depicted.

To provide an explanation of the above code, a minimum amplitude atwhich points will be defined (min_amp) and a maximum amplitude at whichpoints will be defined (max_amp) may be provided. A number of points tobe used to define the waveform in the MATLAB program (numpts) may thenbe defined.

The MATLAB program may use the following syntax to define a spline:yy=spline(x,Y,xx)  (5)where x represents the X values of a series of points to be used todefine the spline, and Y represents the equivalent Y values of theseries of points to be used to define the spline. The spline program maythen use a cubic spline interpretation to find yy, the values of thisfunction at the values xx. In this case, the x values of the series ofpoints to be used to define the spline are the starting point, a middlepoint, and the end point, which go from 1 to the “numpoints” value, usedto define the length of the desired waveform. The Y values may bedefined based on the minimum and maximum complex amplitude valuesspecified earlier; for example, the first x point may have an amplitudeof 1, the second x point may have an amplitude of 2, and the third mayhave an amplitude of 1. The values calculated by the cubic splineinterpolation are saved as the variable “amps,” and may be used as theamplitudes for a spiral waveform.

The angles to be used in generating the complex waveform may then begenerated. According to the exemplary embodiment displayed above, thisstep may define a number of angles, in specific increments, that go from0 at the start of the series to 2π at the end of the series (meaningthat a full loop of the spiral is defined).

The spiral waveform may then be generated. As depicted, the effects ofthe not-a-knot condition on the spiral waveform, for example the slightrise in amplitude above the defined values immediately after thestarting point and immediately before the ending point, may be shown inthe plot of exemplary FIG. 1.

According to an exemplary embodiment, a second exemplary technique mayinvolve a second variation of splines, and may include a “ZeroDerivative” end condition, where the first derivative associated witheach of the first and the last points is defined to be zero. Aspiral-based symbol waveform whose intra-symbol amplitude variation isdetermined by the spline “Zero Derivative” end condition may beconstructed, for example, from the following exemplary MATLAB code aswould be understood by a person of ordinary skill in the art.

Exemplary MATLAB code for Spline, “Zero Derivative” end condition:

-   -   min_amp=1; % Minimum complex amplitude    -   max_amp=2; % Maximum complex amplitude    -   numpts=32; % Number of points in symbol waveform    -   x=[1 ceil(numpts/2) numpts]; % Indices for start, mid, end        amplitudes    -   y=[min_amp max_amp min_amp]; % Start, middle, end complex        amplitudes    -   idx=1:numpts; % Indices to interpolate into    -   cs=spline(x, [0 y 0]); % Form interpolation polynomial    -   amps=ppval(cs, idx); % Interpolate amplitudes using spline    -   angles=(idx−1)*2*pi/(numpts-1); % Angles for spiral    -   spiral=amps.*exp(1j*angles); % Spiral=amplitudes.*complex circle

Exemplary FIG. 2 depicts a plot of the resulting spiral-based symbolwaveform and its spectrum usage. A real 202 and an imaginary 204component of the spiral may be depicted.

The spiral waveform defined by the second exemplary technique may begenerated by code broadly similar to the code used in the firstexemplary technique and used to generate the spline having a“not-a-knot” end condition. A notable distinction between the second setof code and the first set of code, however, may be the use of slightlydifferent syntax in the “spline” function used to generate the datapoints used to define the spline. In this second exemplary embodiment,the MATLAB program may use the following syntax to define a spline:cs=spline(x,Y)  (6)wherein the cs value is the piecewise polynomial form of the cubicspline. In this case, the y value may be given as the vector [0 y 0];according to the MATLAB syntax, this may cause the first and last value(both zero) to be specified as the endslopes/first derivatives for thecubic spline, establishing the “zero derivative” end condition. The codethen may evaluate the piecewise polynomial, using the ppval function todo so, to generate a number of values that may be saved as the variable“amps,” and which may be used as the amplitudes for a spiral waveform.The code may otherwise continue as it did in the first exemplaryembodiment.

A third exemplary technique may involve the use of linear amplitudevariation, or linear growth/decay (e.g. Archimedes Spiral). For example,according to an exemplary embodiment, a spiral-based symbol waveformwhose intra-symbol amplitude variation is linear (known as an“Archimedes Spiral”) may be governed, in polar coordinate form, by theequationr=aθ  (7)wherein a is a constant. In Cartesian coordinate form, this equationwould becomex ² +y ² =a ²[arctan(y/x)]².  (8)

An Archimedes Spiral may also be constructed from the followingexemplary MATLAB code, as would be understood by a person of ordinaryskill in the art. In this implementation, the amplitude may be ramped uplinearly for the first half of the cycle (which would correspond to onehalf of the symbol time), then down for the other half of the cycle.More sophisticated implementations based on the idea of an ArchimedesSpiral may be possible and covered by the present invention which mayavoid a sharp amplitude derivative discontinuity at the peak.

Exemplary MATLAB code for Linear Growth/Decay (Archimedes Spiral):

-   -   min_amp=1; % Minimum complex amplitude    -   max_amp=2; % Maximum complex amplitude    -   numpts=32; % Number of points in symbol waveform    -   idx=1:numpts; % Indices to interpolate into    -   up_slope=(max_amp−min_amp)/floor((numpts/2)−1);    -   down_slope=−up_slope;    -   up_vec=min_amp+up_slope.*(idx(1:floor(numpts/2)))−1);    -   down_vec=max_amp+down_slope.*(idx(1:floor(numpts/2)))−1);    -   amps=horzcat(up_vec, down_vec); % Linear up then linear down        amplitudes    -   angles=(idx−1)*2*pi/(numpts−1); % Angles for spiral    -   spiral=amps.*exp(1j*angles); % Spiral=amplitudes.*complex circle

Exemplary FIG. 3 depicts a plot of the resulting spiral-based symbolwaveform and its spectrum usage. A real 302 and an imaginary 304component of the spiral may be depicted.

To provide an explanation of the above code, a minimum amplitude atwhich points will be defined (min_amp) and a maximum amplitude at whichpoints will be defined (max_amp) may be provided. A number of points tobe used to define the waveform in the MATLAB program (numpts) may thenbe defined.

The program may then define the upward slope that the spiral-basedwaveform will have as it approaches the peak (up_slope) and the downwardslope that the spiral-based waveform will have as it moves away from thepeak (down_slope). In this case, as the code defines a spiral-basedsymbol waveform whose intra-symbol amplitude variation is linear, theupward slope and the downward slope may each be constant. Vectors ofpoints of the spiral-based waveform as it approaches the peak (up_vec)and as it moves away from the peak (down_vec) may then each becalculated. These vectors may then be saved as the variable “amps,” andmay be used as the amplitudes for a spiral waveform.

The angles to be used in generating the complex waveform may then begenerated. According to the exemplary embodiment displayed above, thisstep may define a number of angles, in specific increments, that go from0 at the start of the series to 2π at the end of the series (meaningthat a full loop of the spiral is defined).

This may yield the following equation for an Archimedes Spiral functionin complex space:

$\begin{matrix}{{f(t)} = {\left( {a + {bt}} \right)e^{\frac{2\;\pi\; j}{C}t}}} & (9)\end{matrix}$wherein a is a constant (a starting amplitude), b is a slope, C is aconstant used to define the rate of angular progression of the spiral incomplex space, and j is an imaginary square root of minus one. Thisequation may be used to generate an exemplary embodiment of a waveformused for communication. According to an exemplary embodiment, thewaveform may grow and shrink in magnitude according to the ArchimedesSpiral equation; for example, according to an exemplary embodiment, awaveform may have positive slope for half of the values of time t and anegative slope for the other half of the values of time t.

The foregoing description and accompanying drawings illustrate theprinciples, preferred embodiments and modes of operation of theinvention. However, the invention should not be construed as beinglimited to the particular embodiments discussed above. Additionalvariations of the embodiments discussed above will be appreciated bythose skilled in the art.

Therefore, the above-described embodiments should be regarded asillustrative rather than restrictive. Accordingly, it should beappreciated that variations to those embodiments can be made by thoseskilled in the art without departing from the scope of the invention asdefined by the following claims.

What is claimed is:
 1. A method for communicating, comprising: obtainingat least one input communication symbol selected from a set ofcommunication symbols; converting the at least one input communicationsymbol into at least one transmittable waveform using at least onenon-periodic function selected from a set of non-periodic functions; andtransmitting the at least one transmittable waveform over acommunication channel, wherein at least one non-periodic functionselected from the set of non-periodic functions is a complex circlefunction multiplied by a piecewise function, the piecewise functioncomprising a plurality of cubic piecewise polynomials and having theform ${S(x)} = \left\{ \begin{matrix}{{{p_{1}(x)} = {a_{1} + \;{b_{1}x} + {c_{1}x^{2}} + {d_{1}x^{3}}}},{x \in \left\lbrack {x_{0},x_{1}} \right\rbrack}} \\\vdots \\{{{p_{n}(x)} = {a_{n} + {b_{n}x} + {c_{n}x^{2}} + {d_{n}x^{3}}}},{x \in \left\lbrack {x_{n - 1},x_{n}} \right\rbrack}}\end{matrix} \right.$ wherein, for a cubic polynomial p defined over aparticular interval, a, b, c, and d are constants defined for that cubicpolynomial over the particular interval, and n is an integer equal to orgreater than 2, wherein the piecewise function is generated from asystem of equations comprising a plurality of boundary conditions and atleast one spline function, the spline function having the form:${{\frac{x_{k} - x_{k + 1}}{6}f_{k - 1}^{''}} + {\frac{x_{k + 1} - x_{k + 1}}{3}f_{k}^{''}} + {\frac{x_{k + 1} - x_{k}}{6}f_{k + 1}^{''}}} = {\frac{f_{k + 1} - f_{k}}{x_{k + 1} - x_{k}} - \frac{f_{k} - f_{k - 1}}{x_{k} - x_{k - 1}}}$wherein k is an integer in a set of integers having a first point k₀ anda last point k_(n), x_(k) is a tabulated data value of x at a particularvalue of k, ƒ_(k) is a tabulated value of a function ƒ at a particularvalue of k, and ƒ_(k)″ is a tabulated value of the second derivative ofa function ƒ at a particular value of k.
 2. The method of communicatingof claim 1, wherein the plurality of boundary conditions comprises a“not-a-knot” condition set, wherein the third derivative of ƒ_(k) whenk=k₁ is continuous and the third derivative of ƒ_(k) when k=k_(n−1) iscontinuous.
 3. The method of communicating of claim 1, wherein theplurality of boundary conditions comprises a clamped spline endcondition set, wherein the first derivative of ƒ_(k) when k=k₀ isspecified and the first derivative of ƒ_(k) when k=k_(n) is specified.4. The method of communicating of claim 3, wherein the first derivativeof ƒ_(k) when k=k₀ is specified to be zero and the first derivative ofƒ_(k) when k=k_(n) is specified to be zero.